WHAT DOES EUCLID HAVE TO SAY ABOUT THE FOUNDATIONS OF COMPUTER SCIENCE? D. E. STEVENSON Abstract. It is fair to say that Euclid's Elements has b een a driving factor in the developmentof mathematics and mathematical logic for twenty-three centuries. The author's own love a air with mathematics and logic start with the Elements. But who was the man? Do we know anything ab out his life? What ab out his times? The contemp orary view of Euclid is much di erent than the man presented in older histories. And what would Euclid say ab out the status of computer science with its rules ab out everything? 1. Motivations Many older mathematicians mayhave started their love a air with mathematics with a high scho ol course in plane geometry. Until recently, plane geometry was taught from translations of the text suchasby Heath [5]. In myown case, it was the neat, clean picture presented with its neat, clean pro ofs. I would guess that over the twenty-three hundred year history of the text, Euclid must be the all-time, b est selling mathematics textb o ok author. At the turn of the last century, David Hilb ert also found a love of Euclid. So much so that Hilb ert was moved to prop ose his Formalist program for mathematics. Grossly over-simpli ed, Hilb ert prop osed that mathematics was a \meaningless game" in that the rules of logic preordained the results indep endent of any interpretation. Hilb ert was rather vehemently opp osed by L. E. J. Brouwer. Brouwer prop osed that it was the intuition of the mathematician who used a p ersonal language to develop mathematics; this language always has meaning [14 ]. A fo cus of Brouwer was the concept of existence. Pretending to 1991 Mathematics Subject Classi cation. 14A99. 1 use an ob ject that has not b een constructed is called impredication. Henr Poincar e said impredi- cation was the ro ot of all mathematical evil. Hilb ert's program eventually would fail, done in by Godel's Incompleteness Theorem. But by no means did this establish intuitionism as the wave of the future. Intuitionism has several facets; it is the existence no impredication facet that broughtmeto study absolute geometry part of Bo ok I with an eye to the history and philosophy of mathematics in Euclidean times. The cause for this was an article byvon Plato [15 ] who prop osed constructive 1 axioms for geometry Brie y, the di erence between constructive and platonic can be stated in terms of Plato and his student Aristotle. This story is well-known in philosophy. Hop efully, not oversimplifying, the question is over existence. Platonic mathematics p ostulates that things just exist. For example, the uncountable set of real numb ers exists as an ideal and we can treat them as if we had access to this set. Essentially, this is a statement of Plato's philosophy. Plato's most famous pupil, Aristotle, disagreed, saying that things must be shown to exist by some metho d or other. Aristotle's stand is often called \empiricism." In mo dern garb, the argument was between Hilb ert's formalism and Brouwer's intuitionism. It is said that \All mathematicians are intuitionists on Sunday and Platonist the rest of the week." Errett Bishop is the most recent prop onent[1]. I nd the intuitionistic argument far more satisfying in computer science: you cannot compute with something until you have created it. According to the the Platonist, Euclid was a platonic mathematician | go gure. This view was p opularized primarily by Pro clus who once headed Plato's Academy and wrote an extensive commentary on the Elements[5 ]. Unfortunately, Pro clus lived ab out six hundred years after Euclid. The history of the platonic versus constructionist views are well do cumented in [2 , 6 ] and the details 1 I use the terms intuitionism and constructivism interchangeably. This may not b e technically correct but the term construction is an inherent part of the geometric discussion 2 Alexander of Aphro disias Amphinomus Anaxgoras Antiphon Aristotle Bryson Demo citus Dio cles Eudemus Euto cius of Aokolon Hero Nicomedes Pappus Plutarch Plato Pro clus Protagoras Simplicius Theon of Smyrna Figure 1. Ma jor Commentators on Geometry are outside our story other than to say conventional wisdom would have said that the constructionist view was not Euclid's. The received story of Euclid is a jumble of comments collected in many contexts. To hear the philosophers tell it, Euclid was as much a philosopher as mathematician. Many logicians talked of a grand scheme by Euclid to set the foundations of mathematics and mathematical logic. But was this his goal? And what was the intellectual climate preceding and during Euclid's time even though we do not know precisely when that was? There is confusion over what commentators have read into the text and what the ancient geometers might themselves have b elieved. Our problem was to understand the \true" Euclid. Section 2 lo oks at the men and their times. Section 3 lo oks at the text of the Elements itself. What might b e Euclid's view of how geometry is conducted in Section 4. 2. The Ancients There seem to be far more ancient commentators Fig. 1 on geometry than there are investigators Fig 2. The purp ose of this section is to identify those p eople who probably had a direct in uence on the technical content of the Elements. 3 Mid 5th Century to Mid 4th Century Hipp o crates of Chios Oenipides of Chios Eratosthenes of Cyrene Time of Plato and Aristotle Archytas of Tarentum Dinostratus Eudoxus of Cnidus Leo damus Leon Menaechmus Theaetetus of Athens Post Euclid App olonius Archimedes Figure 2. Ma jor Contributors to Geometry 2.1. The Men Themselves. Of all those mentioned in the history of geometry there are four pre-Euclideans whom I will mention here. There are two groups: one in the 4th Century and the other in the time of Plato in the 3rd Century. 2.1.1. 4th Century Investigators. In the 4th century BC, Hipp o crates and Oenopides gure promi- nently in the century preceding the establishment of Plato's Academy. Oenopides is reputed to b e the p erson who required only rulers and compasses b e used. In several places, Knorr [9 ] expresses doubt that such a requirement was ever made. It seems highly unlikely that the early geometers put any arti cial imp ediments in the way of development[9,p. 345]. Hipp o crates was, apparently, the geometer of the 4th century. He added signi cantly to the knowledge of his time, but he was also considered the rst compiler of geometry [7, p. 179]. 4 2.1.2. Plato's Academy. After Hipp o crates and Oenopides there is a lull in geometry research but there is also the founding of Plato's Academy. Plato himself was not a mathematician and the Academywas not a hotb ed of mathematical research. Plato is known to have commented that the metho ds of geometry were not abstract enough; several centuries later, Carpus would complain that the metho ds were to o abstract to b e useful [9 , p. 364{365]. The Academy did make a signi cant turn towards mathematics when Eudoxus joined it. Eudoxus. Eudoxus is most rememb ered for his work on approximations to limits, or the so-called \metho d of exhaustion." He was also a consummate geometer. How much did he know ab out constructive metho ds? There is a group of historians that place heavy emphasis on religious motives for solving geometric problems. Seidenb erg [11 , 12 ] calls these \p eg and cord" metho ds. These techniques were known throughout Western Asia. The story is that the priests of various religions solved several imp ortant problems constructively and the Greek mathematicians to ok it from there. Interestingly enough, Eudoxus did sp end time in Egypt studying astronomy with the priests. The apparent impredication in Theorem 1, Bo ok 1 can be seen di erently through the metho ds discussed in Seidenb erg [11 , 12 ]. One need not invoke a continuity principle at all, but simply observe the use of cord and peg metho ds. On the other hand, religion was probably just another application [9 , p. 16]. Eudoxus [6] is given credit for developing deductive reasoning. He also seems to have made a distinction b etween magnitudes and numb ers. While numb ers were prior and ruled by \common notions", magnitudes were not. Magnitudes were ruled by ratios and prop ortions. Menaechmus. Menaechmus was at the Academy at the same time as Eudoxus, probably as a student, then as a teacher. Menaechmus puts the priorityto problems over theorems [9, p. 76, p. 351]. There are twotyp es of problems [9 , p. 359]: A. Those that lead to the determination by certain gures that result in another gure and B. Those that lead to the construction of a gure with sp eci ed prop erties. 5 Name Dates So crates 469 BC to 399 BC Plato c. 428 BC to c. 347 BC Eudoxus 408 BC to 355 BC Aristotle 384 BC to 322 BC Euclid No rm guesses: 325 BC to 300 BC Archimedes 287 BC to 212 BC Theon c. 335 AD to 395 AD Pro clus 410 AD to 485 AD Figure 3. Chronology of Imp ortant Greeks Menaechmus seems to b e the rst recorded intuitionist fo cusing on establishing existence. Euclid. Much controversy could be avoided if we knew more ab out Euclid the man.